Answer :
Answer:
The probability is 0.0014
Step-by-step explanation:
When we have two possible outcomes p, q and we repeat the experiment multiple times, we apply the Binomial distribution, in this distribution the probability of getting exactly k successes in n trials is given by
[tex]P(X=k) = nCk(p)^k(1-p)^{n-k}[/tex]
where p represents the probability of success. nCk refers to the combinations of k elements out of n elements.
In this case, p= .29 (the probability that the person says "yes" being confident that passenger trips to the moon will occur in their lifetime.
q = .71 (The probability that the person says "no")
We sample 200 americans, therefore, we have n = 200 and we need the probability that less than 40% of the people sampled answer "yes".
The 40% of 200 is 80. So, in other words, we need less than 80 people to answer yes to the question.
P(x<80) = P(x=0) + P(x=1) + P(x=2) + .... + P(x=79) = 0.00143466294 (we can use a Binomial Distribution calculator to compute this result)
Therefore, P(x < 80) = 0.0014
Using the normal distribution and the central limit theorem, it is found that there is a 0.9997 = 99.97% probability that less than 40% of the people sampled will answer yes to the question.
Normal Probability Distribution
In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- It measures how many standard deviations the measure is from the mean.
- After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.
- By the Central Limit Theorem, for a proportion p in a sample of size n, the sampling distribution of sample proportion is approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1 - p)}{n}}[/tex], as long as [tex]np \geq 10[/tex] and [tex]n(1 - p) \geq 10[/tex].
In this problem:
- Twenty-nine percent of Americans say they are confident that passenger trips to the moon will occur in their lifetime, hence p = 0.29.
- 200 Americans are sampled, hence n = 200.
The mean and the standard error are given by:
[tex]\mu = p = 0.29[/tex]
[tex]s = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.29(0.71)}{200}} = 0.0321[/tex]
The probability that less than 40% of the people sampled will answer yes to the question is the p-value of Z when X = 0.4, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.4 - 0.29}{0.0321}[/tex]
[tex]Z = 3.43[/tex]
[tex]Z = 3.43[/tex] has a p-value of 0.9997.
0.9997 = 99.97% probability that less than 40% of the people sampled will answer yes to the question.
To learn more about the normal distribution and the central limit theorem, you can take a look at https://brainly.com/question/24663213